That won't work at all. In particular, [itex]f = \sqrt{a^2}[/itex] is incorrect and [itex]df/da[/itex] does not exist.

The most straightforward way to compute this derivative is to simply write out the function f. You recall that [itex]|x + iy| = \sqrt{x^2 + y^2}[/itex], right? Apply the definition of modulus, and you should get something you could do back in calc I.